J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449--467, 1965.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....known for undirected graphs The problem of finding a 3 cycle cover in undirected graphs can be solved in polynomial time by Tutte s reduction [15] to the classical perfect matching problem in undirected graphs. The classical perfect matching problem can be solved in polynomial time (see Edmonds [6]) The corresponding maximization problem can be solved in polynomial time, even if we allow arbitrary nonnegative weights. Hartvigsen [10] has designed a powerful polynomial time algorithm for deciding whether a given undirected graph has a 4 cycle cover. He has also presented a polynomial time ....

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449--467, 1965.

....of perfect matchings of G, i.e. PMA (G) convf j M is a perfect matching of G g: By definition, PMA (G) is a 0=1 polytope. This polytope has been studied extensively and in fact played an essential role in shaping the exciting evolution of combinatorial optimization theory, see Edmonds [2], Schrijver [6, Section 8.10] and Lovasz Plummer [5] We present here two of the most important results on the perfect matching polytope concerning simple H representations of the matching polytope. The first one is for the case of bipartite graphs and the second one for the general nonbipartite ....

J. Edmonds. Path, trees, and flowers. Canadian J. Math., 17:449--467, 1965.

....and only if G has no Tutte set. If G has a dnp matching M, then M is a perfect matching of G # . Conversely, if M is a perfect matching of G # , then M is a dnp matching of G. Given G as input, the graph G # clearly may be constructed in polynomial time. By a well known result of Edmonds [5], checking for a perfect matching is in P, so the recognition problem for ncc graphs is in P. Corollary 5. The following problem is in P. INSTANCE: A graph G. QUESTION: Does G have the ncc property 3. Operations preserving the ncc property Suppose that G and H are two graphs with dnp ....

J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 17 (1965), 449-467.

....only using odd alternating paths, then it is an odd vertex. Free vertices are also even. The sets of even and odd vertices are unique, i.e. they are independent of the particular 1The length of a path is the number of edges it contains. choice of a maximum matching. Edmonds also proved this in [7]. A non reachable vertex is called out of Jbvest vertex . 4.2 The algorithm The data structure we maintain consists of a sparse blossom forest, parity informations (even, odd, or out of forest) for the vertices, and a list consisting of the edges in a current maximum matching. The matching and ....

....edges are marked for quick recognition. Thus, it is trivial to answer a query. Additionally, we store at each node in the blossom forest a pointer to the tree that it belongs to. A blossom forest is a well known data structure used in static maximum cardinality matching algorithms, see, e.g. [7, 20, 30]. We will show below that the union of a maximum matching of the current graph and a blossom forest with respect to it is a suitable subgraph for the current graph. It follows that s(n) O(n) Conceptually, the data structure is a sparse subgraph of the current graph G, which has the same ....

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449 467, 1965.

....order to find a good heuristic solution for the MWMP. 1 Introduction Complexity in Theory and Practice. In the field of discrete algorithms, the classical way to distinguish easy and hard problems is to study their worst case behavior. Ever since Edmonds seminal work on maximum matchings [7, 8], the adjective good for an algorithm has become synonymous with a worst case running time that is bounded by a polynomial in the input size. At the same time, Edmonds method for finding a maximum weight perfect matching in a complete graph with edge weights serves as a prime example for a ....

J. Edmonds. Paths, trees, and flowers. Can. J. Mathematics, 17 (1965), 449--467.

....matching. To our knowledge, nothing is known for values k 3. The problem 3 UCC of finding a 3 cycle cover in undirected graphs can be solved in polynomial time by Tutte s reduction [16] to the classical perfect matching problem in undirected graphs which can be solved in polynomial time (see [3]) Also Min 3 UCC can be solved in polynomial time, even if we allow arbitrary nonnegative weights. Hartvigsen [7] designs a powerful polynomial time algorithm for 4 UCC which can also be applied to solve Min 4 UCC. He also presents a polynomial time algorithms that computes a minimum weight ....

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449--467, 1965.

....integers in UE The type of characterization of Theorem 5. 1 has been obtained for a number of complexity classes (see Immerman [Imm87] and a survey by Immerman [Imm89] The most interesting complexity class to consider is P (deterministic polynomial time) Ever since Cobham [Cob64] and Edmonds [Edm65], the class P has been often identified with the class of feasible problems. An ordered structure is one with a built in linear order , that is, a structure over a language that contains a binary relation symbol , where the interpretation of in the structure is a linear order of the ....

J. Edmonds. Paths, trees, and flowers. Canadian J. Math., 17:449--467, 1965.

....independence number. Results of this type are detailed below. Independence Number. Three different approaches have been taken in producing polynomial algorithms for determining the independence number ff(G) of a graph G. An important result underlying two of the approaches is due to Edmonds in [51], where he exhibited a polynomial algorithm for finding a maximal matching in a graph. A key idea that was used in [51] was the notion of an augmenting path, which is, given a matching M , a path whose edges are alternately in M and E(G) Gamma M , and whose end vertices are not incident to any ....

....been taken in producing polynomial algorithms for determining the independence number ff(G) of a graph G. An important result underlying two of the approaches is due to Edmonds in [51] where he exhibited a polynomial algorithm for finding a maximal matching in a graph. A key idea that was used in [51] was the notion of an augmenting path, which is, given a matching M , a path whose edges are alternately in M and E(G) Gamma M , and whose end vertices are not incident to any edge of M . When going to the line graph of a graph, a maximum matching is transformed into a maximum independent set. ....

[Article contains additional citation context not shown here]

Edmonds, J.: Paths, Trees, and Flowers. Canad. J. Math. 17(1965) 449-467

....ihre Anwendungen of the DFG (Deutsche Forschungsgemeinschaft) 1 paths. There is a long history of cardinality matching algorithms, see for example [LP86] but in the following we mention only a few of them, especially those used for computational comparisons later on in this paper. Edmonds [Edm65] obtained the first polynomial time algorithm for this problem. Gabow [Gab76] and Lawler [Law76] showed how to implement Edmonds algorithm to run in O(n 3 ) Using disjoint set union the running time can be reduced to O(nmff(n; m) or further to O(nm) using the linear time set union algorithm ....

Jack Edmonds, Paths, trees, and flowers, Can. J. Math. 17 (1965), 449--467.

....is a big gap between these two problems. The maximum weight stable set problem is NP hard, even if w(v) 1 for v 2 V (see [8] On the other hand, many polynomial time algorithms for the maximum weight cardinality matching problem have been proposed, e.g. 11, 9, 5] for bipartite graphs and [2, 3, 15, 6] for general graphs. Moreover, these polynomial time algorithms have been extended to those solving more general problems, for instance, the maximum weight cardinality stable set problem for claw free graphs [18, 16, 14] the linear matroid parity problem [12, 13, 7, 17] and the linear ....

J. Edmonds, Paths, trees, and flowers, Canad. J. Math. 17 (1965), 449-467.

....to search for augmenting paths. But until 1965, only exponential algorithms for finding a maximum cardinality matching in nonbipartite graphs were known. The reason was that one did not know how to treat odd cycles, the socalled blossoms in alternating 1 paths. In his pioneering work, Edmonds [7] solved this problem by shrinking these odd cycles. In [2, 10, 16, 19] it is shown how to avoid explicit shrinking of odd cycles. All these algorithms need O(n 3 ) or O(nm) time, where n is the number of nodes, and m is the number of edges in the graph. The first polynomial algorithm for the ....

Edmonds J., Paths, trees, and flowers, Canad. J. Math, 17 (1965), 449-- 467.

....is one of the classic problems of algorithmic graph theory. Applications of the maximum matching problem can be found in many textbooks, such as [10] Several algorithms have been proposed to solve the maximum matching problem on general graphs. The first polynomial algorithm was found by Edmonds [4]. The complexity of Edmonds algorithm has been improved from O(n 4 ) to O(n 3 ) by Gabow [6] and using Union and Find [7] to O(mn) The algorithm with the best asymptotic running time O( p nm) is the algorithm of Micali and Vazirani [14] Unfortunately the algorithm of Micali and Vazirani ....

J. Edmonds. Paths, tress, and flowers. Canad. J. Math., 17:449--467, 1965.

....optimization problems. It means that an adjacency criterion for a class of polyhedra could provide a basis for an efficient algorithm which uses some sort of local search technique. In fact, there exist such algorithms, e.g. various algorithms for minimum cost perfect matching problems [4, 7, 10, 11], and the successful heuristic for traveling salesman problems proposed by Lin and Kernighan [19] Some efficient algorithms for ranking problems are also based on adjacency criteria for some combinatorial polytopes [8, 12] However, it may be very difficult to test adjacency for a class of ....

J. Edmonds, Path, trees, and flowers, Canadian J. Math. 17 (1965) 449--467.

....search infeasible, came into focus only much later. A theory of efficient computability emerged in the early sixties, with the notion of polynomial time computation 4 , as a theoretical benchmark of efficiency, crystallizing in combinatorial optimization theory, especially in Edmonds s work [Ed], as well as in the Soviet school of mathematical cybernetics (cf. Tr] By no means should one interpret a polynomial time algorithm as a necessarily efficient one in a practical sense. Nevertheless, the positive side of this theory turned out to be quite successful: by outlawing exhaustive ....

Edmonds, J.: Path, trees, and flowers. Canadian J. of Math. 17 (1965), 449-- 467.

....with the vertex v: The variable x(e) denotes the number of times the edge e is traversed in the postman s tour. The Chinese postman problem is a well solved problem [8] and actually Edmonds presented a polynomial time algorithm by transforming the problem into a non bipartite matching problem [5, 6]. 3 Department of Management Science, Science University of Tokyo, 1 3 Kagurazaka, Shinjuku ku, Tokyo 162, Japan. saru ms.kagu.sut.ac.jp or saruwata gssm.otsuka.tsukuba.ac.jp y Department of Industrial Administration, Science University of Tokyo, 2641 Yamazaki, Nodashi, Chiba 278, Japan. ....

....of the above algorithm. In each iteration, we delete one CPP from the set of problems P and add at most two CPPs to P ; i.e. the number of problems in the set P increases at most 1. Hence, the memory requirement of the algorithm is less than O(K(n m) By applying Edmonds technique in [5], the ordinary Chinese postman problem is reduced to a non bipartite matching problem and we can obtain a matching type optimal solution of CPP(G;w;a; b; V 1 ) in polynomial time [1, 3, 4] Here we denote the computational efforts required to obtain a matching type optimal solution in Step 1 by ....

J. Edmonds, Path, trees, and flowers, Canadian J. Math., 17 (1965), pp. 449--467.

....some combinatorial optimization problems. It means that an adjacency criterion for a class of polyhedra could provide a basis for an efficient algorithm which uses some sorts of local search technique. In fact, there exist such algorithms, e.g. Edmonds blossom algorithm for matching problems [7] and the successful heuristic for traveling salesman problems proposed by Lin and Kernighan [14] However, it may be very difficult to test adjacency for some combinatorial polyhedra. In [17] Papadimitriou showed that it is NP complete to decide whether two given vertices of a symmetric traveling ....

J. Edmonds, Path, trees, and flowers, Canadian J. Math. 17 (1965) 449--467.

....some free vertex is of even length 3 , then v is an even vertex. If v is reachable, but only using odd alternating paths, then it is an odd vertex. Free vertices are also even. The sets of even and odd vertices are unique, i.e. they are independent of the particular choice of a maximum matching [7]. A non reachable vertex is called out of forest vertex. 8.2 Data Structure and Suit(G) The data structure we maintain consists of a sparse blossom forest, parity informations (even, odd, or out of forest) for the vertices, and a list consisting of the edges in a current maximum matching. The ....

....The matching and forest edges are marked. Thus, it is trivial to answer a query. Additionally, we store at each node in the blossom forest a pointer to the tree that it belongs to. A blossom forest is a well known data structure used in static maximum cardinality matching algorithms, see, e.g. [7, 23, 33]. Conceptually, the data structure is a sparse subgraph of the current graph G, which has the same matching number and the same parities as G. Even, odd, and out of forest vertices correspond to the Gallai Edmonds Decomposition of a graph. For a definition and properties of this decomposition 3 ....

[Article contains additional citation context not shown here]

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449 -- 467, 1965.

No context found.

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449--467, 1965.

No context found.

Edmonds J., Paths, trees, and flowers, Canad. J. Math, 17 (1965), 449--

No context found.

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449--467, 1965.

No context found.

J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449--467, 1965.

No context found.

Edmonds J., Paths, trees, and flowers, Canad. J. Math, 17 (1965), 449-- 467.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC